Summary and InterpretationLet's review what we've learned from the meanfield solution to the Ising Model.
Interpreting the MeanField SolutionThe optimized value of has a very nice and particular physical interpretation. Let me write down the selfconsistency expression again: The LHS is easy to interpret: it's the ‘‘effective magnetic field’’ felt by some spin in the lattice under the trial Hamiltonian. In other words, it's the value of the magnetic field that we pretend each spin experiences when we write down the trial Hamiltonian. The RHS is the mean field experienced by spin , and our selfconsistency relation says that there's two contributions to the mean field: one is the external field applied to all the spins in the model, and the other is the mean field from all of 's neighbors. To see why this is true, rewrite the RHS as where is the average magnetization of a trialHamiltonianspin. Here represents the average magnetic field created by a spin with average magnetization . If you are surrounded by nearest neighbors, and you feel a field of from each of your neighbors, then you feel a total field of from your neighbors. So the selfconsistency relation tells you that: Interpretation of b
In the trial Hamiltonian, we pretend that each spin experiences an ‘‘effective’’ magnetic field , resulting from the external field as well as the field set up by perfectlyaverageneighbors. (In reality, though, neighbors aren't always perfectly average…) Notice that the average magnetization of these neighbors is in determined by the effective value of the field …which is must be determined by the average magnetization …so it's a selfconsistent sort of relation: the mean field is exactly the right strength to induce a magnetization that produces exactly that same mean field. (Think about it.) Is mean field theory accurate?From the interpretation above, we've learned that mean field theory does successfully introduce interactions, but in a rather peculiar and naive way: rather than assuming that each spin actually interacts with its neighbors, it assumes that each spin interacts with an abstract ‘‘superneighbor’’ that behaves like a perfectly averaged mean magnetization of the entire material. In reality, of course, the neighbors of any particular spin don't behave perfectly like their idealaveragemagnetizationsuperneighbor; rather, they're all jiggling around from thermal energy all the time, which means that the actual field experienced by any particular spin will fluctuate over time. For instance, one particular spin could be very unlucky, and 5 out of its 6 neighbors might all happen to point upwards – and then the field that it actually feels will be much more ‘‘up’’ than the material's mean field. So the main drawback to mean field theory is that it does not properly account for fluctuations in the microenvironment (is that even the right word?) around any particular spin. Washing out fluctuations with high dimensionWell, if fluctuations are the nailintheshoe that ruin the validity of mean field theory, we should expect that the less fluctuations there are in a particular site's meanfield, the more accurate mean field theory is. And as we remember from statistics, the more things we average over, the less the average is going to fluctuate around the mean. Let me say that again in a box: Life Lesson
The more things you average over, the less it fluctuates around the mean. (To be more precise, the standard deviation of the average of lots of independent random variables goes like . The reason is that their variances add up linearly as , and we take a square root to get the standard deviation of the sum as , and then we divide by to get the std.dev. of the average as . It's always nice to review your stats…) What this means for us is that the more neighbors you interact with, the closer mean field theory lies to the truth. Typically, what this means is that the dimensionality of your system has to be high enough, because the number of neighbors scales with . So the higher the dimensionality, the more accurate the mean field, and in fact, above a certain upper critical dimension, the results of mean field theory become exact. (Or so I've heard. I haven't actually understood the proof of this yet…) Mean Field Theory fails in 1DAs an example of how poorly meanfield theory can behave, let's consider again the 1D Ising model, which we found the exact solution for last week. There, we figured out that the magnetization stays at all the way down to zero temperature. In other words, there is no phase transition in the 1D Ising model. On the other hand, if you treat the 1D Ising model with the meanfield approach, it will predict that the magnetization becomes nonzero once the temperature is cold enough ()! This is clearly an incorrect prediction of meanfield theory.
Another ‘‘issue’’ common to all meanfield theories is that they incorrectly predict the correlations between spins. Remember that in the exact 1D Ising model solution, we found that there was a correlation between nearby spins, where if a particular spin happened to point up, then a nearby spin would probably point up as well (for ). More precisely, the 2site correlation was given by , which decays exponentially with a characteristic lengthscale as the distance between the spins increased. On the other hand, mean field theory fails to predict this behavior, because the trial Hamiltonian has a noninteractig form where each of the spins lives in its own energetically isolated world…and remember that there can be no correlations without interaction terms in the Hamiltonian, because the partition function just factorizes. TakeawayIn summary, we've learned that Mean field theory is better in high dimensions, because you have more neighbors to average over. To address some of the concerns of mean field theory, we will go over a more advanced theory, called LandauGinzburg theory, in the coming weeks. Click here to begin! Leave a Comment Below!Comment Box is loading comments...
